def get_invariant_qf(c_lst_lst):
'''
Parameters
----------
c_lst_lst : list
A list of "c_lst"-lists.
A c_lst is a list of length 9 with elements
c0,...,c8 in the subring in QQ(k) of "MARing.FF".
The matrix
[ c0 c1 c2 ]
M = [ c3 c4 c5 ]
[ c6 c7 c8 ]
represents---for each value of k---an
automorphism of P^2. If we set k:=0 then
"c_lst" must correspond to the identity matrix:
[ 1,0,0, 0,1,0, 0,0,1 ]. If M is not normalized
to have determinant 1 then the method should be
taken with care (see doc. ".get_c_lst_lst_dct").
Returns
-------
list<MARing.R>
A list of quadratic forms in the ideal of the Veronese surface V
(see ".get_ideal_lst()"), such that the quadratic forms are
invariant under the automorphisms of V as defined by "c_lst_lst"
and such that the quadratic forms generate the module of
all invariant quadratic forms. Note that Aut(V)=Aut(P^2).
'''
# for verbose output
#
mt = MATools()
# initialize vectors for indeterminates of "MARing.R"
#
x = MARing.x()[:6]
q = MARing.q()[:6]
r = MARing.r()[:6]
# obtain algebraic conditions on q0,...,q19
# so that the associated quadratic form is invariant
# wrt. the automorphism defined by input "c_lst_lst"
#
iq_lst = []
for c_lst in c_lst_lst:
iq_lst += Veronese.get_invariant_q_lst(c_lst)
iq_lst = list(MARing.R.ideal(iq_lst).groebner_basis())
# solve the ideal defined by "iq_lst"
#
sol_dct = MARing.solve(iq_lst, q)
# substitute the solution in the quadratic form
# associated to the symmetric matrix qmat.
#
qmat = Veronese.get_qmat()
qpol = list(sage_vector(x).row() * qmat *
sage_vector(x).column())[0][0]
sqpol = qpol.subs(sol_dct)
mt.p('sqpol =', sqpol)
mt.p('r =', r)
assert sqpol.subs({ri: 0 for ri in r}) == 0
iqf_lst = [] # iqf=invariant quadratic form
for i in range(len(r)):
coef = sqpol.coefficient(r[i])
if coef != 0:
iqf_lst += [coef]
mt.p('iqf_lst =', iqf_lst)
return iqf_lst
def get_invariant_qf(c_lst_lst, exc_idx_lst=[]):
'''
Computes quadratic forms in the ideal of the
double Segre surface that are invariant under
a given subgroup of Aut(P^1xP^1).
Parameters
----------
c_lst_lst : list<list<MARing.FF>>
A list of "c_lst"-lists.
A c_lst is a list of length 8 with elements
c0,...,c7 in QQ(k),
where QQ(k) is a subfield of "MARing.FF".
If we substitute k:=0 in the entries of
"c_lst" then we should obtain the list:
[1,0,0,1,1,0,0,1].
A c_lst represents a pair of two matrices:
( [ c0 c1 ] [ c4 c5 ] )
( [ c2 c3 ] , [ c6 c7 ] )
with the property that
c0*c3-c1*c2!=0 and c4*c7-c5*c6!=0.
If the two matrices are not normalized
to have determinant 1 then the method should be
taken with care (it should be checked that the
tangent vectors at the identity generate the
correct Lie algebra).
exc_idx_lst : list<int>
A list of integers in [0,8].
Returns
-------
A list of quadratic forms in the ideal of (a projection of)
the double Segre surface S:
".get_ideal_lst( exc_idx_lst )"
such that the quadratic forms are invariant
under the automorphisms of S as defined by "c_lst_lst"
and such that the quadratic forms generate the module of
all invariant quadratic forms. Note that Aut(S)=Aut(P^1xP^1).
'''
# for verbose output
#
mt = MATools()
sage_set_verbose(-1)
# initialize vectors for indeterminates of "MARing.R"
#
x = MARing.x()
q = MARing.q()
r = MARing.r()
# obtain algebraic conditions on q0,...,q19
# so that the associated quadratic form is invariant
# wrt. the automorphism defined by input "c_lst_lst"
#
iq_lst = []
for c_lst in c_lst_lst:
iq_lst += DSegre.get_invariant_q_lst(c_lst, exc_idx_lst)
iq_lst = list(MARing.R.ideal(iq_lst).groebner_basis())
# solve the ideal defined by "iq_lst"
#
sol_dct = MARing.solve(iq_lst, q)
# substitute the solution in the quadratic form
# associated to the symmetric matrix qmat.
#
qmat = DSegre.get_qmat(exc_idx_lst)
qpol = list(sage_vector(x).row() * qmat *
sage_vector(x).column())[0][0]
sqpol = qpol.subs(sol_dct)
mt.p('sqpol =', sqpol)
mt.p('r =', r)
assert sqpol.subs({ri: 0 for ri in r}) == 0
iqf_lst = [] # iqf=invariant quadratic form
for i in range(len(r)):
coef = sqpol.coefficient(r[i])
if coef != 0:
iqf_lst += [coef]
mt.p('iqf_lst =', iqf_lst)
return iqf_lst
def test__get_aut_P8__rotation_matrix(self):
'''
OUTPUT:
- We verify that for the 1-parameter subgroup of rotations
with matrix
[ cos(k) -sin(k) ]
[ sin(k) cos(k) ]
it is for our Lie algebra methods sufficient to consider
1-parameter subgroups that have the same tangent vector
at the identity. Note that for k=0 we need to get the identity
and the determinant should be 1.
'''
k = ring('k')
I = ring('I')
a, b, c, d, e, f, g, h = ring('a, b, c, d, e, f, g, h')
x = MARing.x()
q = MARing.q()
r = MARing.r()
#
# We consider the representation of the
# following matrix into P^8
#
# ( [ 1 -k ] , [ 1 0 ] )
# ( [ k 1 ] [ 0 1 ] )
#
# We define A to be the tangent vector at the
# identity of this representation
#
N = DSegre.get_aut_P8([1, -k, k, 1] + [1, 0, 0, 1])
A = MARing.diff_mat(N, k).subs({k: 0})
#
# We consider the representation of the
# following matrix into P^8
#
# ( [ cos(k) -sin(k) ] , [ 1 0 ] )
# ( [ sin(k) cos(k) ] [ 0 1 ] )
#
# We define B to be the tangent vector at the
# identity of this representation
#
M = DSegre.get_aut_P8([a, b, c, d] + [e, f, g, h])
a, b, c, d, e, f, g, h = sage_var('a, b, c, d, e, f, g, h')
k = sage_var('k')
M = sage__eval(str(list(M)), {
'a': a,
'b': b,
'c': c,
'd': d,
'e': e,
'f': f,
'g': g,
'h': h,
'k': k
})
M = sage_matrix(M)
M = M.subs({
a: sage_cos(k),
b: -sage_sin(k),
c: sage_sin(k),
d: sage_cos(k),
e: 1,
f: 0,
g: 0,
h: 1
})
# differentiate the entries of M wrt. k
dmat = []
for row in M:
drow = []
for col in row:
drow += [sage_diff(col, k)]
dmat += [drow]
M = sage_matrix(dmat)
B = M.subs({k: 0})
assert str(A) == str(B)